Spectral statistics and dynamics of Levy matrices

We study the spectral statistics and dynamics of a random matrix model wher e matrix elements are taken from power-law tailed distributions. Such distr ibutions, labeled by a parameter mu, converge on the Levy basin, giving the matrix model the label "Levy matrix" [P. Cizeau and J. P. Bouchaud, Phys. Rev. E 50, 1810 (1994)]. Such matrices are interesting because their proper ties go beyond the Gaussian universality class and they model many physical ly relevant systems such as spin glasses with dipolar or Ruderman-Kittel-Ka suya-Yosida interactions, electronic systems with power-law decaying intera ctions, and the spectral behavior at the metal insulator transition. Regard ing the density of states we extend previous work to reveal the sparse matr ix limit as mu-->0. Furthermore, we find for 2 x 2 Levy matrices that geome trical level repulsion is not affected by the distribution's broadness. Nev ertheless, essential singularities particular to Levy distributions for sma ll arguments break geometrical repulsion and make it mu dependent. Level dy namics as a function of a symmetry breaking parameter gives new insight int o the phases found by Cizeau and Bouchaud (CB). We map the phase diagram dr awn qualitatively by CB by using the Delta(3) statistic. Finally we compute the conductance of each phase by using the Thouless formula, and find that the mixed phase separating conducting and insulating phases has a unique c haracter. [S1063-651X(99)00910-1].